Enhanced fractional multi-period repetitive controller with fractional time-delay for high-precision servo systems

Repetitive control (RC) is widely used for achieving high-precision tracking and effective rejection of periodic disturbances in servo and vibration-prone systems. Conventional single-period and multi-period repetitive controllers, however, rely on integer delay models that cannot exactly represent the true fundamental periods of multi-frequency periodic signals, leading to frequency mismatch and residual steady-state errors. To address this limitation, this paper proposes a Fractional Multi-Period Repetitive Controller (FMPRC) that preserves non-integer delay lengths within the discrete-time multi-period internal model, thereby maintaining accurate frequency alignment between the controller and actual periodic components. The total delay is decomposed into integer and fractional components, where the fractional delay is realized using either infinite impulse response (IIR) or finite impulse response (FIR) filters. Unlike conventional delay rounding approaches, the proposed formulation embeds fractional-delay modeling directly into the internal model while preserving unity gain, causality, and closed-loop stability under the plug-in RC framework. Stability conditions and practical implementation aspects of the FMPRC are analyzed, and both IIR- and FIR-based realizations are systematically compared. Simulation studies on high-precision servo systems subject to multi-frequency and time-varying periodic disturbances demonstrate that the proposed FMPRC significantly improves steady-state tracking accuracy and periodic disturbance rejection compared to conventional single-period and multi-period repetitive controllers. These results demonstrate the effectiveness of the proposed FMPRC as multiple-frequency vibration suppression strategy for high-precision motion and servo control systems.